Smmetry 

(Under construction)

Definition

 Symmetry is a fundamental organizing principle in nature and in culture. The analysis of symmetry allows for understanding the organization of a pattern, and provides a means for determining both invariance and change. 
  A pattern, whether in nature or art, relies upon three characteristics: a  unit, repetition, and a system of organization.
In art, symmetry can be considered as a  geometrical duality.
In theoretical physics, symmetry is sort of complementarity.

Asymmetry: 

Symmetry Breaking: 

POSSIBILITIES FOR THE COMPOSITION of a design are limitless, and may rely upon choices. But possibilities for the repetition of that design, whether symmetrical or asymmetrical, are limited by the laws of pattern formation and are subject to the constraints of symmetry.

IN ALL PATTERNS there are four basic symmetry operations that may be performed upon a fundamental region, design or motif. Mathematicians call these rigid motions because they suggest movements without distortion of size or shape around a point, along or across a line, or to cover a plane.

HERE, THE LETTER F (and the blank space around it) is taken as our fundamental region to demonstrate the four basic symmetry operations or rigid motions:

translation rigid motion with repetition along a line	reflection rigid motion with repetition across a line (axis)
glide reflection rigid motion with reflected repetition along a line	rotation rigid motion with repetition around a point

IN RUG-WEAVING, the repetition of a design to form a pattern is accomplished by counting and repeating sequences of knots. The basic symmetries in carpets are thus effected knot by knot.

IN CARPETS, BORDER PATTERNS result when any or several of the basic symmetries are repeated in one direction. The constraints of symmetry are such that there are seven (7) possible combinations:
 

translation

horizontal reflection

vertical reflection

reflection +  reflection

glide reflection

rotation

reflection +  glide reflection

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FIELD PATTERNS result when symmetrical repetition takes place in two directions, thus forming a two-dimensional pattern that covers the plane. There are seventeen (17) systems which mathematicians classify as symmetry groups:
 

translations	reflections	reflections + reflections
glide reflections	reflections + glide reflections	rotations (2)
reflections + rotations (2)	rotations (2) + glide reflections	rotations (2) + reflections + reflections
rotations (4)	reflections + rotations (4)	rotations (4) + reflections
rotations (3)	reflections + rotations (3)	rotations (3) + reflections

rotations (6)

reflections + rotations (6)

THE EASIEST WAY TO ANALYZE a pattern is to locate points of rotation, and lines of symmetry. Why? Because the rigid motions require centers of rotation and axes of repetition or reflection for symmetry to be present.

WHAT IS AN AXIS? An axis is a visible or implied line that is vertical, horizontal, or diagonal, along which designs are repeated or reflected to form patterns.

WHAT IS A GRID? A grid is a visible or implied series of points, or axes that intersect. Grids underly the structure of all two-dimensional patterns.

square grid	triangular grid	hexagonal grid
rectangular grid	rhomboid grid	oblique grid (square)

Grids are usually based on regular polygons: squares, equilateral triangles, and hexagons. Or they can be based on rectangles, parallelograms and rhomboids.

THE ARRANGEMENT OF POLYGONS that forms a grid is called a tessellation. Other shapes may also tessellate.

WHAT IS A TESSELLATION? A tessellation is a pattern formed by the repetition of a single unit or shape that, when repeated, fills the plane with no gaps and no overlaps. Familiar examples of tessellations are the patterns formed by paving stones or bricks, and cross-sections of beehives.

Tessellations are not typical of Oriental carpets except as visible grid structures. Although they often appear in minor borders, only rarely are tessellations used as field patterns. 
 

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The application of symmetry to physics leads to the important conclusion that certain physical laws, particularly conservation laws, are unaffected by symmetry operations on the geometric coordinates of the particles concerned, including time, when it is considered as a fourth dimension; i.e., the laws remain valid at all places and times in the universe. In particle physics, considerations of symmetry can be used to derive conservation laws and to determine which particle interactions can take place and which cannot (the latter are said to be forbidden). Symmetry also has applications in many other areas of physics and chemistry—for example, in relativity and quantum theory, crystallography, and spectroscopy. Crystals and molecules may indeed be described in terms of the number and type of symmetry operations that can be performed on them. The quantitative discussion of symmetry is called group theory.

Valid symmetry operations are those that can be performed without changing the appearance of an object. The number and type of such operations depends on the geometry of the object to which the operations are applied. The meaning and variety of symmetry operations may be illustrated by considering a square lying on a table. For the square, valid operations are (1) rotation about its centre through 90, 180, 270, or 360 degrees, (2) reflection through mirror planes perpendicular to the table and running either through any two opposite corners of the square or through the midpoints of any two opposing sides, and (3) reflection through a mirror plane in the plane of the table. There are therefore nine symmetry operations that yield a result indistinguishable from the original square. A circle would be said to have higher symmetry because, for example, it could be rotated through an infinite number of angles (not just multiples of 90 degrees) to give an identical circle.

Subatomic particles have various properties and are affected by certain forces that exhibit symmetry. An important property that gives rise to a conservation law is parity. In quantum mechanics all elementary particles and atoms may be described in terms of a wave equation. If this wave equation remains identical after simultaneous reflection of all spatial coordinates of the particle through the origin of the coordinate system, then it is said to have even parity. If such simultaneous reflection results in a wave equation that differs from the original wave equation only in sign, then the particle is said to have odd parity. The overall parity of a collection of particles, such as a molecule, is found to be unchanged with time during physical processes and reactions; this fact is expressed as the law of conservation of parity. At the subatomic level, however, parity is not conserved in reactions, owing to the weak nuclear force responsible for radioactivity.

Elementary particles are also said to have internal symmetry; these symmetries are useful in classifying particles and in leading to selection rules. Such an internal symmetry is baryon number, which is a property of a class of particles called hadrons. Hadrons with a baryon number of zero are called mesons, those with a number of +1 are baryons. By symmetry there exists another class of particles with a baryon number of -1; these are called antibaryons. Baryon number is conserved during nuclear interactions.
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